In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called an arc or line).[1] Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A. In contrast, if any edge from a person A to a person B corresponds to A's admiring B, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called an undirected graph and the edges are called undirected edges while the latter type of graph is called a directed graph and the edges are called directed edges.

In one very common sense of the term,[4] a graph is an ordered pairG = (V, E) comprising a setV of vertices, nodes or points together with a set E of edges, arcs or lines, which are 2-element subsets of V (i.e., an edge is associated with two vertices, and the association takes the form of the unordered pair of the vertices). To avoid ambiguity, this type of graph may be described precisely as undirected and simple.

Other senses of graph stem from different conceptions of the edge set. In one more general conception,[5]E is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, E is a multiset of unordered pairs of (not necessarily distinct) vertices. Many authors call these types of object multigraphs or pseudographs.

All of these variants and others are described more fully below.

The vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge.

V and E are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case. Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set. The order of a graph is |V|, its number of vertices. The size of a graph is |E|, its number of edges. The degree or valency of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends (a loop) is counted twice.

For an edge {x, y}, graph theorists usually use the somewhat shorter notation xy.

The edges E of an undirected graph G induce a symmetric binary relation ~ on V that is called the adjacency relation of G. Specifically, for each edge {x, y}, the vertices x and y are said to be adjacent to one another, which is denoted x ~ y.

As stated above, in different contexts it may be useful to refine the term graph with different degrees of generality. Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise, graph means "undirected simple finite graph" (see the definitions below).

A directed graph.

A simple undirected graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph.

An undirected graph is a graph in which edges have no orientation. The edge (x, y) is identical to the edge (y, x). That is, they are not ordered pairs, but unordered pairs—i.e., sets of two vertices {x, y} (or 2-multisets in the case of loops). The maximum number of edges in an undirected graph without a loop is n(n − 1)/2, where n is the number of nodes in the graph.

A a set of ordered pairs of vertices, called arrows, directed edges (sometimes simply edges with the corresponding set named E instead of A), directed arcs, or directed lines.

An arrow (x, y) is considered to be directed fromxtoy; y is called the head and x is called the tail of the arrow; y is said to be a direct successor of x and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arrow (y, x) is called the inverted arrow of (x, y).

A directed graph G is called symmetric if, for every arrow in G, the corresponding inverted arrow also belongs to G. A symmetric loopless directed graph G = (V, A) is equivalent to a simple undirected graph G′ = (V, E), where the pairs of inverse arrows in A correspond one-to-one with the edges in E; thus the number of edges in G′ is |E| = |A|/2, that is half the number of arrows in G.

An oriented graph is a directed graph in which at most one of (x, y) and (y, x) may be arrows of the graph. That is, it is a directed graph that can be formed as an orientation of an undirected graph. However, some authors use "oriented graph" to mean the same as "directed graph".

A mixed graph is a graph in which some edges may be directed and some may be undirected. It is written as an ordered triple G = (V, E, A) with V, E, and A defined as above. Directed and undirected graphs are special cases.

Multiple edges are two or more edges that connect the same two vertices. A loop is an edge (directed or undirected) that connects a vertex to itself; it may be permitted or not, according to the application. In this context, an edge with two different ends is called a link.

A multigraph, as opposed to a simple graph, is an undirected graph in which multiple edges (and sometimes loops) are allowed.

Where graphs are defined so as to disallow both multiple edges and loops, a multigraph is often defined to mean a graph which can have both multiple edges and loops,[6] although many use the term pseudograph for this meaning.[7] Where graphs are defined so as to allow both multiple edges and loops, a multigraph is often defined to mean a graph without loops.[8]

A simple graph is an undirected graph with neither multiple edges nor loops. In a simple graph the edges form a set (rather than a multiset) and each edge is an unordered pair of distinct vertices. Thus, we can define a simple graph to be a set V of vertices together with a set E of edges, which are 2-element subsets of V.

In a simple graph with n vertices, the degree of every vertex is at most n − 1.

A quiver or multidigraph is a directed multigraph. A quiver may have directed loops in it. Thus, a quiver is a set V of vertices, a set E of edges, and two functions s:E→V{\displaystyle s:E\to V}, t:E→V{\displaystyle t:E\to V}. The map s assigns to each edge its source (or tail), while the map t assigns to each edge its target (or head).

A weighted graph is a graph in which a number (the weight) is assigned to each edge.[9] Such weights might represent for example costs, lengths or capacities, depending on the problem at hand. Some authors call such a graph a network.[10][11]Weighted correlation networks can be defined by soft-thresholding the pairwise correlations among variables (e.g. gene measurements). Such graphs arise in many contexts, for example in shortest path problems such as the traveling salesman problem.

A regular graph is a graph in which each vertex has the same number of neighbours, i.e., every vertex has the same degree. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.

In an undirected graph, an unordered pair of vertices {x, y} is called connected if a path leads from x to y. Otherwise, the unordered pair is called disconnected.

A connected graph is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a disconnected graph.

In a directed graph, an ordered pair of vertices (x, y) is called strongly connected if a directed path leads from x to y. Otherwise, the ordered pair is called weakly connected if an undirected path leads from x to y after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called disconnected.

A strongly connected graph is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a disconnected graph.

A k-vertex-connected graph or k-edge-connected graph is a graph in which no set of k − 1 vertices (respectively, edges) exists that, when removed, disconnects the graph. A k-vertex-connected graph is often called simply a k-connected graph.

A bipartite graph is a graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2.

In a complete bipartite graph, the vertex set is the union of two disjoint sets, W and X, so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X.

A path graph or linear graph of order n ≥ 2 is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the {vi, vi+1} where i = 1, 2, …, n − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph.

A cycle graph or circular graph of order n ≥ 3 is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the {vi, vi+1} where i = 1, 2, …, n − 1, plus the edge {vn, v1}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.

Two edges of a graph are called adjacent if they share a common vertex. Two arrows of a directed graph are called consecutive if the head of the first one is the tail of the second one. Similarly, two vertices are called adjacent if they share a common edge (consecutive if the first one is the tail and the second one is the head of an arrow), in which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident.

The graph with only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges is known as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or empty graph, but the terminology is not consistent and not all mathematicians allow this object.

Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called vertex-labeled. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called edge-labeled. Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled. (Note that in the literature, the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.

^Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 19. ISBN978-0-486-67870-2. Retrieved 8 August 2012. A graph is an object consisting of two sets called its vertex set and its edge set.

1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Travelling salesman problem
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It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science. TSP is a case of the travelling purchaser problem and the vehicle routing problem. In the theory of computational complexity, the version of the TSP belongs to the class of NP-complete problems. Thus, it is possible that the running time for any algorithm for the TSP increases superpolynomially with the number of cities. The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization and it is used as a benchmark for many optimization methods. The TSP has several applications even in its purest formulation, such as planning, logistics, slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. The TSP also appears in astronomy, as observing many sources will want to minimize the time spent moving the telescope between the sources. In many applications, additional constraints such as limited resources or time windows may be imposed, the origins of the travelling salesperson problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, the travelling salesperson problem was mathematically formulated in the 1800s by the Irish mathematician W. R. Hamilton and by the British mathematician Thomas Kirkman. Hamilton’s Icosian Game was a puzzle based on finding a Hamiltonian cycle. Hassler Whitney at Princeton University introduced the name travelling salesman problem soon after, Dantzig, Fulkerson and Johnson, however, speculated that given a near optimal solution we may be able to find optimality or prove optimality by adding a small amount of extra inequalities. They used this idea to solve their initial 49 city problem using a string model and they found they only needed 26 cuts to come to a solution for their 49 city problem. As well as cutting plane methods, Dantzig, Fulkerson and Johnson used branch, in the following decades, the problem was studied by many researchers from mathematics, computer science, chemistry, physics, and other sciences. Christofides made a big advance in this approach of giving an approach for which we know the worst-case scenario and his algorithm given in 1976, at worst is 1.5 times longer than the optimal solution. As the algorithm was so simple and quick, many hoped it would give way to a optimal solution method. However, until 2011 when it was beaten by less than a billionth of a percent, Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP. This supplied a mathematical explanation for the apparent computational difficulty of finding optimal tours, great progress was made in the late 1970s and 1980, when Grötschel, Padberg, Rinaldi and others managed to exactly solve instances with up to 2392 cities, using cutting planes and branch-and-bound. In the 1990s, Applegate, Bixby, Chvátal, and Cook developed the program Concorde that has used in many recent record solutions

3.
Mixed graph
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A mixed graph G = is a mathematical object consisting of a set of vertices V, a set of edges E, and a set of directed edges A. Consider adjacent vertices u, v ∈ V, a directed edge, called an arc, is an edge with an orientation and can be denoted as u v → or. Also, an edge, or edge, is an edge with no orientation. For the purpose of our example we will not be considering loops or multiple edges of mixed graphs. A cycle of a graph, or mixed cycle, is formed if the directed edges of the mixed graph form a cycle. An orientation of a graph is considered acyclic if cycles cannot be formed from the directed edges. We call a mixed graph acyclic if all of its orientations are acyclic, mixed graph coloring can be thought of as a labeling or an assignment of k different colors to the vertices of a mixed graph. Different colors must be assigned to vertices that are connected by an edge, the colors may be represented by the numbers from 1 to k, and for a directed arc, the tail of the arc must be colored by a smaller number than the head of the arc. For example, consider the figure to the right and our available k-colors to color our mixed graph are. Since u and v are connected by an edge, they must receive different colors or labelings and we also have an arc from v to w. Since orientation assigns an ordering, we must label the tail with a smaller color than the head of our arc. A proper k-coloring of a graph is a function c, V → where. Referring back to our example, this means that we can label both the head and tail of with the positive integer 2, a coloring may or may not exist for a mixed graph. In order for a graph to have a k-coloring, the graph cannot contain any directed cycles. If such a k-coloring exists, then we refer to the smallest k needed in order to color our graph as the chromatic number. We can count the number of proper k-colorings as a function of k. This is called the polynomial of our graph G and can be denoted as χ G. The deletion–contraction method can be used to compute weak chromatic polynomials of mixed graphs and this method involves deleting an edge or arc and contracting the remaining vertices incident to that edge to form one vertex

4.
Partition of a set
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In mathematics, a partition of a set is a grouping of the sets elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets. A partition of a set X is a set of nonempty subsets of X such that every element x in X is in one of these subsets. Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold, the union of the sets in P is equal to X. The sets in P are said to cover X, the intersection of any two distinct sets in P is empty. The elements of P are said to be pairwise disjoint, the sets in P are called the blocks, parts or cells of the partition. The rank of P is |X| − |P|, if X is finite, every singleton set has exactly one partition, namely. The empty set ∅ has exactly one partition, namely ∅, for any nonempty set X, P = is a partition of X, called the trivial partition. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, the set has these five partitions, sometimes written 1|2|3. The following are not partitions of, is not a partition because one of its elements is the empty set, is not a partition because the element 2 is contained in more than one block. Is not a partition of because none of its blocks contains 3, however, thus the notions of equivalence relation and partition are essentially equivalent. The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly one element from each part of the partition and this implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class. Informally, this means that α is a fragmentation of ρ. In that case, it is written that α ≤ ρ and this finer-than relation on the set of partitions of X is a partial order. Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, the partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left. These atomic partitions correspond one-for-one with the edges of a complete graph, in this way, the lattice of partitions corresponds to the lattice of flats of the graphic matroid of the complete graph. Another example illustrates the refining of partitions from the perspective of equivalence relations, if D is the set of cards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalence classes, the sets and. The 2-part partition corresponding to ~C has a refinement that yields the same-suit-as relation ~S, which has the four equivalence classes, and. In other words, given distinct numbers a, b, c in N, with a < b < c, if a ~ c, it follows that also a ~ b and b ~ c, that is b is also in C

5.
Connectivity (graph theory)
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It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network, a graph is connected when there is a path between every pair of vertices. In a connected graph, there are no unreachable vertices, a graph that is not connected is disconnected. A graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints, a graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected, in an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are connected by a path of length 1, i. e. by a single edge. A graph is said to be connected if every pair of vertices in the graph is connected, a connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge, a directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected graph. It is connected if it contains a path from u to v or a directed path from v to u for every pair of vertices u, v. It is strongly connected, diconnected, or simply strong if it contains a path from u to v. The strong components are the maximal strongly connected subgraphs, a cut, vertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected. The connectivity or vertex connectivity κ is the size of a minimal vertex cut, a graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater. In particular, a graph with n vertices, denoted Kn, has no vertex cuts at all. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. The local connectivity κ is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs, a graph G which is connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges, in the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More generally, a cut of G is a set of edges whose removal renders the graph disconnected. A graph is called k-edge-connected if its edge connectivity is k or greater, if u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex

6.
James Joseph Sylvester
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James Joseph Sylvester FRS was an English mathematician. He made fundamental contributions to theory, invariant theory, number theory, partition theory. He played a role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University. At his death, he was professor at Oxford, Sylvester was born James Joseph in London, England. His father, Abraham Joseph, was a merchant, at the age of 14, Sylvester was a student of Augustus De Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife, subsequently, he attended the Liverpool Royal Institution. Sylvester began his study of mathematics at St Johns College, Cambridge in 1831, for the same reason, he was unable to compete for a Fellowship or obtain a Smiths prize. In 1838 Sylvester became professor of philosophy at University College London. In 1841, he was awarded a BA and an MA by Trinity College, following his early retirement, Sylvester published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry. In 1872, he received his B. A. and M. A. from Cambridge. In 1876 Sylvester again crossed the Atlantic Ocean to become the professor of mathematics at the new Johns Hopkins University in Baltimore. His salary was $5,000, which he demanded be paid in gold, after negotiation, agreement was reached on a salary that was not paid in gold. In 1878 he founded the American Journal of Mathematics, the only other mathematical journal in the US at that time was the Analyst, which eventually became the Annals of Mathematics. In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University and he held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair. Sylvester invented a number of mathematical terms such as matrix, graph. He coined the term totient for Eulers totient function φ and his collected scientific work fills four volumes. In Discrete geometry he is remembered for Sylvesters Problem and a result on the orchard problem, Sylvester House, a portion of an undergraduate dormitory at Johns Hopkins University, is named in his honor. Several professorships there are named in his honor also, the collected mathematical papers of James Joseph Sylvester, I, New York, AMS Chelsea Publishing, ISBN 978-0-8218-3654-5 Sylvester, James Joseph, Baker, Henry Frederick, ed

7.
Graph of a function
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In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin ⁡ cos ⁡ is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section

8.
Diagram
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A diagram is a symbolic representation of information according to some visualization technique. Diagrams have been used since ancient times, but became prevalent during the Enlightenment. Sometimes, the uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word graph is used as a synonym for diagram. Specific kind of display, This is the genre that shows qualitative data with shapes that are connected by lines, arrows. In science the term is used in both ways, on the other hand, Lowe defined diagrams as specifically abstract graphic portrayals of the subject matter they represent. Or in Halls words diagrams are simplified figures, caricatures in a way and these simplified figures are often based on a set of rules. The basic shape according to White can be characterized in terms of elegance, clarity, ease, pattern, simplicity, elegance is basically determined by whether or not the diagram is the simplest and most fitting solution to a problem. g. Many of these types of diagrams are generated using diagramming software such as Visio. Chart Diagrammatic reasoning Diagrammatology List of graphical methods Mathematical diagram Plot commons, michael Anderson, Peter Cheng, Volker Haarslev. Theory and Application of Diagrams, First International Conference, Diagrams 2000, edinburgh, Scotland, UK, September 1–3,2000. Garcia, M The Diagrams of Architecture

9.
Complete bipartite graph
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Graph theory itself is typically dated as beginning with Leonhard Eulers 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, Llull himself had made similar drawings of complete graphs three centuries earlier. That is, it is a graph such that for every two vertices v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km, n, for any k, K1, k is called a star. All complete bipartite graphs which are trees are stars, the graph K1,3 is called a claw, and is used to define the claw-free graphs. The graph K3,3 is called the utility graph and this usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings, it is impossible to solve without crossings due to the nonplanarity of K3,3. Given a bipartite graph, testing whether it contains a complete bipartite subgraph Ki, a planar graph cannot contain K3,3 as a minor, an outerplanar graph cannot contain K3,2 as a minor. Conversely, every nonplanar graph contains either K3,3 or the complete graph K5 as a minor, Kn, n is a Moore graph and a -cage. The complete bipartite graphs Kn, n and Kn, n+1 have the possible number of edges among all triangle-free graphs with the same number of vertices. The complete bipartite graph Km, n has a vertex covering number of min, the complete bipartite graph Km, n has a maximum independent set of size max. The adjacency matrix of a bipartite graph Km, n has eigenvalues √, −√ and 0, with multiplicity 1,1. The Laplacian matrix of a bipartite graph Km, n has eigenvalues n+m, n, m. A complete bipartite graph Km, n has mn−1 nm−1 spanning trees, a complete bipartite graph Km, n has a maximum matching of size min. A complete bipartite graph Kn, n has a proper n-edge-coloring corresponding to a Latin square, every complete bipartite graph is a modular graph, every triple of vertices has a median that belongs to shortest paths between each pair of vertices

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that …

A set of stamps partitioned into bundles: No stamp is in two bundles, no bundle is empty, and every stamp is in a bundle.

The 52 partitions of a set with 5 elements. A colored region indicates a subset of X, forming a member of the enclosing partition. Uncolored dots indicate single-element subsets. The first shown partition contains five single-element subsets; the last partition contains one subset having five elements.

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called …

All non-isomorphic graphs on 3 vertices and their chromatic polynomials. The empty graph E3 (red) admits a 1-coloring, the others admit no such colorings. The green graph admits 12 colorings with 3 colors.

A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible.